Stochastic Analysis of Image Acquisition, Interpolation and Scale-space Smoothing

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1 Stochastc Analyss of Image Acquston, Interpolaton and Scale-space Smoothng Kalle Åström, Anders Heyden Dept of Mathematcs, Lund Unversty Box 8, S- Lund, Sweden emal: Abstract In the hgh-level operatons of computer vson t s taken for granted that mage features have been relably detected. Ths paper addresses the problem of feature extracton by scale-space methods. There has been a strong development n scale-space theory and ts applcatons to low-level vson n the last couple of years. Scale-space theory for contnuous sgnals s on a frm theoretcal bass. However, dscrete scale-space theory s known to be qute trcky, partcularly for low levels of scale-space smoothng. The paper s based on two key deas: to nvestgate the stochastc propertes of scale-space representatons and to nvestgate the nterplay between dscrete and contnuous mages. These nvestgatons are then used to predct the stochastc propertes of sub-pxel feature detectors. The modelng of mage acquston, mage nterpolaton and scale-space smoothng s dscussed, wth partcular emphass on the nfluence of random errors and the nterplay between the dscrete and contnuous representatons. In dong so, new results are gven on the stochastc propertes of dscrete and contnuous random felds. A new dscrete scale-space theory s also developed. In practce ths approach dffers lttle from the tradtonal approach at coarser scales, but the new formulaton s better suted for the stochastc analyss of sub-pxel feature detectors. The nterpolated mages can then be analyzed ndependently of the poston and spacng of the underlyng dscretsaton grd. Ths leads to smpler analyss of sub-pxel feature detectors. The analyss s llustrated for edge detecton and correlaton. The stochastc model s valdated both by smulatons and by the analyss of real mages. Mathematcs Subject Classfcaton: 68U, 6D5, 6G6 Introducton Ths work s motvated by an attempt to understand how well edges can be located n mages. Study Fgure a, whch contans a dgtal mage taken wth an ordnary CCD-camera. The scene contans a sharp dscontnuty n Ths work has been done wthn then ESPRIT Reactve LTR project 94, CUMULI and the Swedsh Research Councl for Engneerng Scences (TFR), project a b c Fgure : Fgure a llustrate a part of a dgtal mage contanng a hgh contrast edge. Fgure b llustrate the result from a tradtonal edge-detector. Fgure c llustrate the edge as localzed usng the new methodology presented n ths paper. The result s not only the edge but also an estmate of ts localzaton varance.

2 W deal W w v w e h blur D samplng e nose Fgure : Illustraton of the mage acquston model. The orgnal ntensty dstrbuton W deal s blurred by the kernel h h h, correspondng to camera optcs blur h and dgtzaton blur h. The blurred sgnal W s then sampled to form a dscrete sgnal w. Fnally nose s added. The result s the measured dscrete sgnal v w e. contrast called an edge. It s the mage of a whte paper on a black background. Fgure a contans a small secton of that hgh contrast edge. We would lke to know how well that edge can be located. What does the magntude of these errors depend on? A common msconcepton s that the edge can only be located on a pxel level, that the reason we cannot locate the edge exactly s that we have only got a dscrete set of measurements. In fact, most edge detector algorthms [5, 8, 9,, 3,,, 7] as they are called n the mage analyss lterature gve as output a bnary mage where all the edge pxels are marked wthout any estmate of edge localzaton error, see Fgure b. In ths paper we wll try to present a smple model for mage acquston and mage nterpretaton. Ths model can be used to understand what s measured n dgtal mages, how to construct smple feature detectors and how to analyze ther performance. Thus t wll be shown that edges can be detected wth sub-pxel accuracy and the edge localzaton error can be predcted and estmated usng mage data. Fgure c llustrates the edge as localzed usng the new methodology presented n ths paper. The edge localzaton errors are n the order of / th to / th of a pxel. Image acquston s vewed as a composton of blurrng, deal samplng and added nose, smlar to [4]. The dscrete sgnal s analyzed after nterpolaton. Ths makes t possble to detect features on a sub-pxel bass. Averagng, or scale-space smoothng, s used to reduce the effects of nose. To understand feature detecton n ths framework, one has to analyze the effect of nose on nterpolated and smoothed sgnals. In dong so a theory s obtaned that connects the dscrete and contnuous scale-space theores. The paper s organzed as follows. Secton treats the mage acquston model. In Secton 3 a method for mage nterpretaton s proposed. The dscrete scale-space s nduced from the contnuous scale-space theory. Ths solves the problem of usng fne scales and t also smplfes the theory. The man motvaton for our formulaton of scale-space theory s, however, to smplfy the stochastc analyss of low-level feature detectors. The stochastc propertes of the ntensty error feld are dscussed n Secton 4. A short ntroducton to statonary random felds s gven and some mportant results that are relevant for our model are demonstrated. The deas are verfed wth numercal experments on real mages. The sub-pxel edge detector s studed n Sectons 5 and 6. Sub-pxel correlaton s nvestgated n Secton 7. A word should be sad about the methodology of ths paper. The mage acquston process and the propertes of mage features are dffcult, or mpossble, to fully model a pror. Here we make a mathematcal engneerng approach, startng wth some smple assumptons from whch the analyss starts. In the results obtaned, further approxmatons and assumptons wll be made and motvated. These are thus part of the modelng, and make t possble to gve a stochastc analyss of two major problems n computer vson, edge-detecton and correlaton. The valdty of the modelng s tested afterward n smulatons and experments. The results turn out to justfy the theory. Image acquston To model the mage acquston, the ntensty dstrbuton W deal that would be caught by an deal camera s frst affected by aberratons n the optcs of the real camera, e.g. blurrng caused by sphercal aberraton, coma and astgmatsm. Other aberratons deform the mage, lke Petzval feld curvature and dstorton, see []. Such dstorton can typcally be handled by geometrc consderatons n md-level vson and wll not be commented upon here. One way to model camera blur s to convolve the deal ntensty dstrbuton wth a kernel correspondng to the smoothng caused by the camera optcs. Ths process also removes some amount of the hgh spatal frequences. In a vdeo-camera, the blurred mage ntensty dstrbuton s typcally measured by a CCD array. One can thnk of each pxel ntensty as the weghted mean of the ntensty dstrbuton n a wndow around the deal pxel poston. Takng the weghted mean around a poston s equvalent to frst convolutng wth the weghtng kernel

3 .5 h H h H h.5 H.5.5 Fgure 3: The fgure llustrates the kernels h, h and h correspondng Fourer transforms. h h, representng blurrng and wndowng, and ther and then deal samplng. Ths s llustrated n Fgure. The deal ntensty dstrbuton W deal s frst blurred to form the smooth dstrbuton W. Ths s sampled to form the dscrete sgnal w. Fnally, due to quantzaton and other errors, stochastc errors are ntroduced n the measured dscrete sgnal v. The estmaton of the orgnal contnuous ntensty dstrbuton W deal usng only the dscretzed and nosy mage v s a severely ll-posed nverse problem, that has to be regularsed. The followng assumptons and notatons wll be used. Assumpton.. The blur caused by camera optcs can be modeled as convoluton wth a kernel h, and the wndow blur caused by weghted samplng, modeled as convoluton wth kernel h, where both operatons are of low-pass type. Denote by h the convoluton of these two kernels, h h h, where denotes convoluton. Thus W W deal h. In the defnton of the Fourer transform, we wll use the formula F W f n W τ e π f τ dτ () where f τ denotes the scalar product. Assumpton.. All energy n the hgh spatal frequences s cancelled before dscretzaton. The functon W W deal h s band-lmted,.e. the Fourer transform F W s zero outsde a bounded nterval. The Assumpton. says that we have no alasng effects, when the functon W s sampled. In the sequel the functon W wll be sampled at nteger postons. To avod alasng effects we wll assume that W s band-lmted wthn frequency nterval n. Ths makes t possble to reconstruct W from the sampled data, as wll be dscussed later. For ths purpose we ntroduce the class of functons B n W L n suppf W n () where L denotes the class of square ntegrable functons,.e. functons W such that W x dx. Assumpton.3. The error can be modeled by the addton, after dscretzaton, of a statonary, dscrete random feld, defned n Secton 4. Expermentally t s verfed that the errors n ndvdual pxel ntenstes often can be modeled as ndependent random varables wth smlar dstrbuton. Alasng Some nterestng questons are: What s a reasonable model of the blur caused by camera optcs and dscretzaton,.e. what are the functons h and h above? Are hgh frequences suppressed before dscretzaton? A crude 3

4 sn model of the camera and dscretzaton blur wll be presented, to obtan a feelng for how hgh frequences are attenuated before dscretzaton. Ths s motvated n the one-dmensonal case. The kernels n the general case can be obtaned by tensor multplcaton. The dscusson that follows only serve the purpose of dscussng and justfyng Assumpton.. The exact form of h and h below s not mportant and wll not be used n the sequel. The camera blur or pont-spread functon as t s also called s dscussed n some detal n []. In a well corrected system t can be modeled by the Ary rradance dstrbuton, see [, p. 485]. Dstortons n the camera make t reasonable to model the camera blur as the convoluton wth a Gaussan kernel wth some wdth a,.e. h t (3) The correspondng Fourer transform s πa e t a (4) H f e π a f The measurement or dscretzaton kernel at each pxel s modeled wth a rectangular kernel, h t θ t θ t θ x f x, f x. (5) where θ denotes the Heavsde functon The Fourer transform of h s where snc s the functon H f snc f πs snc s πs f s, f s. The correspondng nterpolaton operator s denoted Thus the total blur caused by optcs and dscretzaton s modeled as a convoluton wth the kernel h h h whose Fourer transform s H H H The kernels and ther transforms are llustrated n Fgure 3 where the parameter a n (3) has the value 6. The result s clearly a flter of low-pass type. To quantfy ths we use the squared norms: A hgh A low H d f H d f H d f These quanttes represents how the flter h attenuates energy at hgh and low spatal frequences. For a wdth of a 6 pxels, we have A low 4 and A hgh 3. Thus the energy n the low frequences s roughly halved whle the energy n the hgh frequences s attenuated by a factor of. To avod alasng, hgh frequency components should be attenuated before dscretzaton. Notce that ths s done to a reasonable extent by the camera. Alasng effects are thus substantally reduced by the normal pre-smoothng n the camera system. The effect s stll larger when the kernel wdth a s ncreased, as s shown n the Table. Thus pre-flterng before samplng s nherent n a camera system. The way t s done depends crtcally on the propertes of each ndvdual camera and the resoluton of the dgtzer. To get the best effect, the camera blur and the dscretzaton grd should be chosen carefully to match each other. Ths s analogous to the desgn of ant-alasng flters n sgnal processng, see [4, p. 468]. Notce n partcular that too sharp an mage wll gve alas effects whch cannot be removed by the subsequent sgnal processng. 4

5 a A low A hgh Table : The table llustrates the attenuaton of low versus hgh spatal frequences for dfferent levels of camera blur. 3 Samplng, nterpolaton and smoothng To formalze the mage acquston dscusson of the prevous secton, some notaton wll be ntroduced. Let upper case letters, W, denote sgnals wth contnuous parameters, whereas lower case letters, w, denote dscrete sgnals. Here, and often n the sequel, we use the word sgnal synonymously wth functon, and dscrete sgnal synonymously wth sequence or functon defned on n, for some n. Defnton 3.. The dscretsaton operator, or samplng operator, D : B w DW W where l denotes square summable sequences,.e. sequences such that n w. Note that the samplng operator maps a contnuous sgnal W onto a dscrete sgnal w. By assumptons. and.3, mage acquston s modeled as a composton of convoluton, deal dscretzaton and addton of nose: W deal blur W h h W deal n l, s samplng w DW nose v w e (7) These assumptons wll serve as an ntal model. Further mprovements can be made by a more detaled camera acquston model. Nevertheless, these assumptons wll help us to model and analyze the next stage, namely estmatng the contnuous mage ntensty dstrbuton from the dscrete mage. Obvously, t s mpossble to reconstruct the orgnal ntensty dstrbuton W deal wthout some a pror knowledge. Compare wth blnd deconvoluton n whch W deal s estmated usng the assumpton that the convoluton kernel has compact support, [7]. It s, however, reasonable to try to estmate the blurred and dstorted ntensty dstrbuton or to estmate an even more blurred verson. Scale-space smoothng W h h W deal h W deal Scale-space theory and ts applcaton to computer vson s dscussed brefly n ths secton. A more thorough treatment s gven n [8]. The dea s to assocate to each sgnal f : n a famly T t f t of gradually smoother sgnals. The orgnal sgnal corresponds to the scale t and ncreasng the scale t should smplfy the mage wthout creatng spurous structure. Each such sgnal captures the behavour of the sgnal at one scale. Ths s llustrated n Fgure 4. It s natural to vew mage acquston as a process that removes fne detals. It wll therefore be dffcult (mpossble) to reconstruct the whole scale-space representaton of W deal. It s however possble to recontruct the sgnal at courser scales. Scale-space s therefore a natural tool for mage nterpretaton. Smoothng s also useful n order to attenuate hgh-frequency nose wthout dsturbng the low-frequency components of the sgnal. There s a trade-off n choosng the smoothng parameter. The real strength n usng the scale-space approach s the possblty to study the whole scale-space representaton, Ths wll, however, not be pursued n ths paper. The emphass wll be made to study the stochastc propertes of each scale-space representaton separately. In the contnuous case, smoothng wth the Gaussan kernel G b x πb e x b 5 (6) (8)

6 t Fgure 4: Scale-space representaton of a one-dmensonal sgnal. The sgnal at dfferent scales can be thought to represent the sgnal at dfferent levels of resoluton or detal. s very natural. In fact f we want to defne a scale-space representaton T t f of a one-dmensonal sgnal f wth the followng propertes: T t s a lnear and translaton nvarant operator for every t, Scale nvarance. If the sgnal s enlarged by some factor λ,.e. g x t t t λ such that T t g x T t f x λ, f x λ then there exsts a scale Generalzed Sem-group property: T t T t f T t Postvty preservng: f T t f, then the only choce s to defne T t f accordng to t f, T t f f G t cf. [4]. Alternate defntons and proofs can be found n [3, 5, 8, 8]. Here we wll use an alternatve scale parameter b t. Defnton 3.. The smoothng operator S b represents convoluton wth the Gaussan kernel G b. A sgnal W s represented at scale b by ts smoothed verson W b : W b S b W G b W (9) The sgnal W b s called the scale-space representaton of W, at scale b. In the sequel subscrpts are used to denote dfferent scales. Ths scale-space representaton has several advantages. It can be shown that structure decrease as scale parameter ncrease n the sense that local extrema are never enhanced,.e. t T t f x f x s a maxmum and t T t f x f x s a mnmum. Another nce feature s that the smoothed functon W b has contnuous dervatves of arbtrary order. A thrd useful property s that the hgh frequency components of the nose are attenuated as scale ncreases. By usng multdmensonal Gaussans, there s a natural generalzaton to functons W of several varables. Scale-space theory n the dscrete tme case has been nvestgated n [8]. It turns out that just by samplng a contnuous scale-space kernel, one obtans a dscrete scale-space kernel. Samplng of the Gaussan kernel can thus be used to obtan a dscrete scale-space kernel. However, n dong so one does not obtan a scale-space theory wth all the nce features of the contnuous scale-space theory. 6

7 There are dffcultes wth fne scales. In partcular t s dffcult to defne hgher order dervatves at fne scale levels. For the same reason t s dffcult to defne local extremum and zero crossngs for fne scales. The sem-group property s lost. These questons are dscussed n [8]. Interpolaton and scale-space smoothng The man dea of our approach s to nduce the dscrete sgnal, the scale spaces, etc. from the assocated nterpolated quanttes. By an nterpolaton or restoraton method we mean an operator that maps a dscrete sgnal, w, to a contnuous one, W. The deal nterpolaton operator s of specal nterest: Defnton 3.3. Ideal nterpolaton I : l w W L s defned by W s Iw s snc s w () We propose to use deal nterpolaton I, and dscretzaton D as mappngs between the contnuous and dscrete sgnals to solve the restoraton and dscrete scale-space problems. In other words we relate the dscrete and contnuous sgnals through the operatons of dscretzaton and deal low-pass nterpolaton. Ths s llustrated by the dagram: W I D w () where D s the dscretzaton operator and I s the deal nterpolaton operator. Note that f the camera nduced blur cancels the hgh frequency components n W as n Assumpton., the determnstc restoraton W s equal to W,.e. W W. Ths can be stated more precsely usng the samplng theorem. Theorem 3.. A contnuous tme sgnal W wth a Fourer transform wth support wthn the nterval s gven unquely by sampled sgnal w D W. The contnuous tme sgnal s obtaned by deal nterpolaton W I w. Proofs and further readng can be found n [6,, 6]. Thus deal nterpolaton s the pseudo-nverse of dscretsaton,.e. D DID () I IDI (3) Usng these defntons, the dscrete and contnuous scale-space representatons can be defned smultaneously and consstently. We propose the followng:. If the prmary nterest s the nterpolated contnuous sgnal, then restore the scale-space smoothed contnuous sgnal W b from the dscrete sgnal w frst usng deal nterpolaton and then contnuous scale-space smoothng.. If the prmary nterest s a dscrete scale-space representaton, then use the nduced representaton from the contnuous scale-space, as defned n (). The procedure s llustrated by the dagram: n W S b W b I w D s b w b (4) 7

8 Thus, from the dscrete sgnal w, the contnuous scale-space smoothed sgnal W b s obtaned as as W b S b I w The dscrete scale-space sgnal w b s b w, s nduced from the contnuous scale-space sgnal,.e. w b s b w def D S b I w (5) where s b s ntroduced as the dscrete scale-space smoothng operator. Notce that s b s a convoluton wth a kernel g b, g b D G b snc (6) The dfferences between ths approach and others, lke the sampled Gaussan approach, s very small for large scales but sgnfcant for small scales. Ths fact s quantfed n the followng lemma. Lemma 3.. For large scale parameter b, snc G b s approxmately equal to G b, n the sense that snc G b G b b π Φ πb (7) where Ph denotes the one-dmensonal normal cumulatve dstrbuton functon. Proof. Consder the Fourer transforms and F G b f e π b f F snc G b χ F G b These are almost equal because F G b s small outsde the nterval when b s large. By Parseval s Theorem we have snc G b G b χ F G b F G b f e π b f d f b p Φ πb (8) The man motvaton for usng deal low-pass nterpolaton s, however, that the approach s well suted for stochastc analyss as wll be shown later. Observe that the nterpolated sgnal W s smooth. Therefore, there s no dffculty n defnng hgher order dervatves. Ths scale-space theory has several theoretcal advantages:. It works for all scales.. The sem-group property, s as b s a b, holds. 3. The couplng to contnuous scale-space theory gves a natural way to nterpolate n the dscrete space. 4. There are no dffcultes n defnng dervatves at arbtrary scales. 5. It s possble to calculate dervatves at arbtrary nterpolated postons. 6. Operators whch commute n the contnuous theory automatcally commute n the dscrete theory. 7. The effect of addtve statonary nose can easly be modeled. 8. It makes t possble to compare the real ntensty dstrbuton wth the nterpolated dstrbuton. There s, however, a prce to pay. The dscrete scale-space smoothng operator s b s a convoluton wth the dscrete functon g b D snc G b.e. s b w g b s b. The dscrete scale-space kernels g b for some values of b are llustrated n Fgure 5. In practce ths scale-space theory s dffcult to use for small scale parameters, because of the large tal of the snc functon. However, the functon snc G b has a very small tal for larger scales. Ths makes t easy to mplement. In practse one may use the approxmaton snc G b G b for large scales, accordng to Lemma 3.. Ths smplfes mplementaton substantally. 8

9 8 8 b b 8 8 b b b b Fgure 5: The dscrete scale-space kernels g b D snc G b for dfferent scales b. Comparson wth other approaches Dscrete scale-space kernels of slghtly dfferent types have been used n the lterature. In ths secton, two of these wll be compared to our approach. Defnton 3.4. The sampled Gaussan dscrete scale-space kernel s defned as g b j j πb e b Defnton 3.5. The ntegrated Gaussan dscrete scale-space kernel s defned as j g b j dx j πb e x b It s well known that these approaches are dffcult to use for fne scales. The dscrete scale-space theores can be understood n the lght of nterpolaton, smoothng and dscretsaton. We wll try to analyse a choce of dscrete scale-space smoothng s b as a composton of some nterpolaton I F, Gaussan smoothng S b and dscretsaton D. We wll use nterpolatons of type W s I F w s F s w (9) where I F s the nterpolaton operator usng dstrbuton F. The queston s then what type of nterpolaton I F corresponds to sampled Gaussan and nterpolated Gaussan,.e. How should F be chosen so that s b DS b I F : W S b W b I F w D s b w b () It s straght forward to see that the operator s b s n fact dscrete convoluton wth a kernel g b D G b F. Example 3.. Dscrete scale space usng the sampled Gaussan s equvalent to nterpolaton wth δ-dstrbuton, followed by Gaussan smoothng and dscretzaton. s b D S b I δ g b D G b δ Ths works well for large scales, but a poor approxmaton s obtaned for small scales as s shown n Fgure 6. 9

10 b b 4 b 6 Fgure 6: Illustraton of nterpolaton usng the sampled Gaussan functon. In the fgure the sgnals W b (dashed) and S b I δ w (full) s shown for b, 4 and 6 The approxmaton S b I δ w W b s poor, for small scale parameter b. The approxmaton s, however, good for large scale parameter b b b 4 Fgure 7: Illustraton of nterpolaton usng the ntegrated Gaussan functon. In the fgure the sgnals W b (dashed) and S b I step w (full) s shown for b, and 4 The approxmaton S b I step w W b s poor, for small scale parameter b. The approxmaton s, however, good for large scale parameter. Defnton 3.6. Pecewse constant nterpolaton s defned as nterpolaton wth the functon. s step s s Example 3.. Dscrete scale space usng the ntegrated Gaussan s equvalent to pece-wse constant nterpolaton, followed by Gaussan smoothng and dscretzaton,.e. s b D S b I step g b D G b step Although t gves better results than the sampled Gaussan t performs poorly at fne scales, cf. Fgure 7. Example 3.3. The proposed scale-space restoraton s made by frst dong deal low-pass nterpolaton and then scale-space smoothng for contnuous sgnals. Ths s llustrated n Fgure 8. Notce the good approxmaton already at scale. Compare wth Fgures 6 and 7. Remark. Observe that the reason why restoraton n general, and restoraton usng the snc functon n partcular, works, s that the sampled functon W has hgh regularty. All hgh frequency components have been attenuated by camera and dgtzaton blur. 4 The random feld model In ths secton the stochastc models are nvestgated. So far we have the followng model for mage acquston W deal blur W D w nose v w e ()

11 b b b 4 Fgure 8: Illustraton of deal low-pass nterpolaton. The fgure llustrate the sgnals W b (dashed) and S b I snc w (full) for b, and 4 The approxmaton S b I snc w W b s good even for small scale parameter b. (In fact the two curves are ndstngushable n the fgure.) where W deal s the deal ntensty dstrbuton enterng the camera system. Ths ntensty s frst blurred. The result W s assumed to have no frequency components outsde the nterval, see Assumpton.. Ths sgnal s then dscretsed and fnally perturbed by addtve nose. The dscrete mage v w e s analyzed drectly or through scale-space smoothng, as llustrated by the dagram: I W E S b w e s b W b E b D w b e b () Note that all operatons are lnear. The stochastc and determnstc propertes can, therefore, be studed separately and the fnal result s obtaned by superposton. Thus wth an a pror model on W deal, for example an deal edge or corner, t s possble to predct the determnstc parts W b and w b. The stochastc propertes of the error felds e, e b, E and E b, wll now be studed. Statonary random felds The theory of random felds s a smple and powerful way to model nose n sgnals and mages. Statonary or wde sense statonary random felds are partcularly easy to use. Denote by E the expectaton value of a random varable. Defnton 4.. A random feld X t wth t n s called statonary or wde sense statonary, f ts mean m t m X t E X t s constant and f ts covarance functon r X t t E X t m t X t m t only depends on the the dfference τ t t. Ths should be compared wth the noton of strct statonarty. Defnton 4.. A random feld X t wth t n s called strctly statonary f for all t t n and all τ the stochastc varable X t X t n has the same probablty dstrbuton as X t τ X t n τ. For statonary felds we wll use r X s t and r X s t nterchangeably as the covarance functon. The analogous defnton s used for a statonary feld n dscrete parameters. The noton of spectral densty R X f F r X f r X τ e π f τ dτ (3) s also mportant. Agan the same defnton can be used for random felds wth dscrete parameters s n, but whereas the spectral densty for random felds wth contnuous parameters s defned for all frequences f, the spectral densty of dscrete random felds R X f r X k e π f k (4) s only defned on an nterval f n. Introductons to the theory of random processes and random felds are gven n [, 6, 7]. These also contans proofs and comments to the followng two useful theorems.

12 Theorem 4. (Convoluton of a random feld). Let X be a statonary random feld and let Y h X, wth h L. Then Y s a statonary random feld wth covarance r Y τ h u h v r X τ u v dudv (5) u v If H s the Fourer transform of h, then the spectral densty functon R Y s R Y f R X f H f (6) Theorem 4. (Samplng of a random feld). If X t s a statonary random feld wth contnuous parameter t n and f x s the result of samplng X at tmes t n,.e. x D X, then x s a statonary dscrete random feld, wth covarance and spectral densty Consder the dagram: R x f E S b E b r x Dr X (7) R X f k (8) k n I e D It follows from the Theorems 4. and 4. that the operators D, S b and s b preserve statonarty. We wll now show that the deal nterpolaton I preserves statonarty as well. Frst we wll analyze the one-dmensonal case. To do ths we need a lemma concernng an nfnte seres: Lemma 4.. s b e b (9) snc s snc t snc s t (3) Proof. The proof follows from a smple calculatons and a formula for summaton of a standard seres see [, p. 88]. Hence, πcotπz lm m snc s snc t sn π s sn π t π s t sn πs sn πt π m n m sn πs sn πt π s t s t sn πs sn πt π z n s t t sn πs sn πt π π s t cot πt cot πs sn πs cos πt sn πt cos πs π s t sn π s t snc s t π s t s t s (3)

13 Ths lemma wll now be used n the proof of the followng theorem, whch descrbes the stochastc propertes of the restored sgnal at scale zero. In the theorem l p s used to denote sequences w such that w p. Theorem 4.3 (Interpolaton of a random process). If e s a statonary dscrete stochastc process wth zero mean and covarance functon r e j r e j such that then the deal nterpolaton at scale zero, r e l p for some p E s snc s e (3) s a well defned random process, wth convergence n quadratc mean. Moreover, E s statonary wth covarance functon r E τ I r e τ r e k snc τ k (33) k Proof. To prove that E s s well defned we need to prove that snc s e converges n the quadratc mean for every s. Let S m snc s e m Convergence n quadratc mean can be establshed usng the Cauchy crteron by showng that E S m S n as m n Here E S m S n n n Ths tends to zero as m n m j m snc s snc s j E e e j n n m j m f the double sum snc s j snc s j r e j snc s snc s j r e j (34) s absolutely convergent. Makng the change of varables, k j l s, the double sum can be rewrtten as j k j l s snc s snc s j r e j snc l snc k l r e k k r e k snc l snc k l (35) l The second sum s the dscrete convoluton of snc s and snc s. Both sequences le n l p for every p. Snce, by Young s nequalty, see [], the convoluton of two functons of type l p and l q s l r wth p q r, the convoluton s of type l r for every r p p, wth p. Hence the sequence f gven by f k snc l snc k l l belongs to l p for every p. Hölder s nequalty then gves that r e k f k (36) k s absolutely convergent f r e l q for some q. It now follows that m E s E snc s e snc s E e. To prove that E s s statonary we need to prove that the covarance r E s t only depends on the dfference s t. The covarance of E s and E t s gven by r E s t E E s E t snc s snc t j E e e j j j k snc s snc t j r e j k r e k snc s snc t k k j snc s snc t k r e k r e k snc s t k I r e s t (37) 3

14 where we have used Lemma 4. to obtan the last but one equalty. Thus the contnuous random process E s s statonary wth covarance functon as descrbed. Remark. The covarance functon s smooth,.e. nfntely dfferentable. Therefore the random feld s C n quadratc mean. The correspondng theorem n hgher dmensons can be proved n exactly the same manner. Theorem 4.4 (Interpolaton of a random feld). Let e n be a statonary dscrete random feld wth zero mean and covarance functon such that r e n j j n r e j n j n r e l p for some p Then the deal nterpolaton (Defnton 3.3) of the dscrete random feld, E I e (38) s a well defned random feld n quadratc mean and E s statonary wth covarance functon r E τ I r e τ (39) Thus, all operatons n the commutatve dagram (9) preserve statonarty. Ths smplfes the modelng of errors n scale-space theory. The effects of the operators I, D, S b and s b on covarance r and spectral densty R are all known by now. It s often convenent to assume that the dscrete nose e can be modeled as whte nose,.e. σ f k, r e k f k. Theorem 4.5. Assume that the dscrete mage v has been obtaned by deal samplng of the blurred mage W wth added whte nose e,.e. v D W e Also assume that supp F W n. Defne the restored ntensty dstrbuton at scale b accordng to V b I v G b where I denotes deal nterpolaton and G b s the Gaussan kernel G b s πb e s b Then V b can be wrtten as V b W G b E b W b E b where E b s a statonary random feld wth covarance functon (4) r Eb snc G b Remark. The theorem states that t s possble to estmate the orgnal contnuous dstrbuton W at scale b usng our methodology. The error E b n ths estmate s a statonary random feld wth known covarance functon. Proof. It follows from the lnearty of nterpolaton and convoluton that V b G b I v G b I w G b I e It follows from Theorem 3. that I w W. The nterpolated error feld E has covarance functon r E τ σ snc τ 4

15 at scale accordng to Theorem 4.4. Usng Theorem 4. and the fact that G b s symmetrc, t follows that r Eb τ σ u G b u G b v snc τ u v dvdu v σ u G b u G b snc τ u du G b G b snc τ G b G b snc τ G b snc τ (4) Remark. The restored mage ntensty dstrbuton V b s a sum of a determnstc part W b and a statonary random feld E b. Notce that the restoraton and the resdual are nvarant of the poston of the dscretzaton grd. The effect of dscretzaton s thus removed. One advantage of the dea of restorng a contnuous scale-space representaton usng the dscrete mage s that t enables us to calculate hgh order dervatves of the mage at arbtrary postons and at any scale. Local features can thus be defned n precse mathematcal terms, and ther poston can be calculated wth a hgh degree of numercal precson. The man motvaton of our formulaton s, however, that t smplfes the analyss of the stochastc propertes of the feature locaton. To do ths we must know the stochastc propertes of dervatves of wde-sense statonary random felds. Ths problem s solved n standard texts on random felds such as []. We have: Theorem 4.6 (Partal dervatves of a random feld). Let X be a statonary random feld wth twce dfferentable covarance functon r X. Then X s dfferentable n mean squares sense. If Y t X s a partal dervatve of X, then Y s also a statonary random feld wth covarance The spectral densty functon R Y s r Y τ t r X τ (4) R Y f R X f π f (43) 5 One-dmensonal edge detecton Stochastc analyss of sub-pxel edge detectors, s one applcaton of ths theory. One-dmensonal edge detecton s treated n ths secton. The problem we want to attack s the followng: Problem 5.. (One-dmensonal edge detecton). Let W deal be a one-dmensonal sgnal whch s smooth except for a fnte set of step-dscontnutes. Let v D h W deal e be the result after smoothng wth kernel h, dscretzaton and added nose. Estmate the poston of the dscontnuty and the uncertanty n ths estmate, usng the dscrete sgnal v. The analyss below can be extended to any functon W deal. For smplcty, t wll be modeled as a deal step functon,.e. as the Heavsde functon, cf. (5). Accordng to the prevous sectons, the scale-space analyss s nvarant to the poston of the dscretzaton grd. Thus, wthout loss of generalty, we can assume that the dscontnuty s at poston. The kernel h s assumed to fulfll Assumpton., and to be approxmately a Gaussan of wdth a, h G a (44) Here, we have mplctly assumed that a s large. The nose wll be modeled as dscrete whte nose wth standard devaton ε. Some of the results that holds exactly usng deal nterpolaton, can be shown to hold approxmately usng the sampled Gaussan approach, for larger scales, see Lemma 3.. The followng sub-pxel edge detector wll be analyzed. Defnton 5. (Edge detecton). Usng the dscrete sgnal v as defned n Problem 5.. Defne edge postons as ponts were the magntude of the dervatve attans a local maxmum. x V b x S b I v 5

16 a W deal d 4 W b 5 b 5 5 w e 5 5 W b 5 c 5 5 W b f 5 5 W b Fgure 9: Illustraton of one-dmensonal sub-pxel edge detecton, usng scale-space smoothng. a: The deal step edge. b: The dscretzed mage wthout nose w (o) and wth nose v w e (+). c-f: The scale space nterpolatons at scale b and ther frst three dervatves. The determnstc sgnal s shown as full lnes, whereas the perturbed sgnal s shown as a broken lne. The sub-pxel edge poston s defned as the poston of the maxmum of the frst dervatve. As ponted out n the prevous secton the problem can be decomposed nto a determnstc part and a random part. The calculatons are then used to predct the stochastc propertes of one-dmensonal edge detectors. The results are then verfed both wth smulated and real data. Determnstc part Frst consder the determnstc part of mage acquston. The gray level profle that hts the CCD-array can be modelled as a gray-level functon. x W deal x A θ x (45) where θ denotes the Heavsde functon, cf. (5). Accordng to the mage acquston model the measured determnstc sgnal s Usng (44) we get w D h W deal DS a W deal : w D h W deal (46) w A Φ a (47) where Φ a s the cumulatve dstrbuton functon of the normal probablty densty functon wth varance a. The dscrete sgnal w s analyzed wth nterpolaton and scale-space smoothng. Ths gves the contnuous sgnal W b S b I w It was assumed that W has no energy outsde the frequency nterval. Thus we have W b S b W S b h W deal S b S a W deal S c W deal, where c a b. Remark. The approxmaton W b W deal G c holds for large b even f we use the sampled Gaussan approach. Ths can be justfed usng a Remann sum as an approxmaton of an ntegral. W b x w G b x W deal G a G b x W deal G a G b x W deal G a G b x W deal G c x (48) 6

17 Agan notce that after scale-space nterpolaton of the dscretzed mage w, we obtan the same result as by smoothng the orgnal ntensty W deal dstrbuton wth G c. The result s nvarant of the poston of the dscretzaton grd. In the followng analyss we wll need W b and ts frst three dervatves: Note that W b x W G c x A Φ c x d (49) W b x A G c x d (5) W b x A x d c G c x d (5) W b x A whch mples that W b has a local extremum at x to determne the stablty of ths local extrema. Random part c x d c 4 G c x d (5) W b d (53) d. We wll need the quantty W b d A c 3 π In ths subsecton we wll derve the effect of addtve random nose to the sgnal. The nose e, s modeled as a whte nose Gaussan process wth mean and varance ε. That s e N ε (55) ε f j After scale space nterpolaton we obtan C e e j (54) f (56) j E b S b I e (57) Accordng to Theorem 4.5 ths s a statonary process wth wth zero mean and covarance functon r Eb τ ε snc G b (58) Accordng to Lemma 3., for coarse scales ths can be approxmated as r Eb τ ε snc G b ε b π e τ 4b accordng to Lemma 3.. Remark. If we use the sampled Gaussan scale-space approach, the approxmaton can be justfed wth explct calculatons. r Eb τ ε τ b π e 4b C s t E X s E X s X t E X t E b π e s b e j b π e t j b π e s b πb e t j b C e e j b π e s b b π e t b ε ε πb e s t 4b e s t b (59) j b e j (6) 7

18 As expected, the sampled Gaussan approach gves an error process E b whch s non-statonary. However, f b s large we can approxmate the last term b b s Θ b e b t e (a Jacob theta-functon) and regard t as a Remann sum of the ntegral, s t s t Θ b b e z s t b s t b dz b π (6) The value of the ntegral s obtaned by varable substtuton. The functon Θ x b s perodc n x wth perod one. The maxmum s obtaned for x and the mnmum for x. The approxmaton s reasonable for b 5 and good for b. We have Combnng (6) and (6) gves C s t ε πb e s Θ x 5 b π 7 Θ x b π 9 Θ x 5 b π t 4b 4 b π ε b π e s t 4b (6) Ths s a covarance functon of a statonary random process, E b, snce t only depends on s t, cf. (59). Scale-space nterpolaton of the dscrete random process s a statonary random process. The statonarty means that the nterpolated error process s ndependent of the underlyng dscretzaton grd. Usng Theorem 4.6 the covarance functon of the frst three dervatves of E b are gven by r E b r E b r E b t re b t t r v E b t t r v E b t where r E t C E s E s t, cf. []. Calculatng the dervatves of r E t ε b π e t 4b (64) gves r E b 4b r E t ε b π e t 4b r E t ε b π e t t ε π e t 4b 5 6b 7 The varances whch are needed later are gven by Observe that the random felds of E b, E b are ndependent, cf. []. r E b 3 8b 5 4b b 9t 3 8b 7 t ε 4b 3 π r E ε 3 b 8b 5 π r E b and E b ε 5 6b 7 π 8b 5 t 5 64b t4 3b 9t4 8b 3t6 all have zero mean. It can also be shown that E (63) (65) (66) b and E b 8

19 Analyss of one-dmensonal edges The determnstc and stochastc analyss wll now be used to analyze the dstrbuton the edge-locaton n the onedmensonal case. As before the measured sgnal v s modeled as a sum of a determnstc part w and a random error process e,.e. v w e The edge poston s defned as the poston where the scale-space nterpolaton V b S b I v has maxmal dervatve,.e. as the zero crossng of V b x. Agan, due to the nvarancy of ths approach, we can assume, wthout loss of generalty, that the true edge s located at d. Usng (53) and (54), the undsturbed second order dervatve of the edge, W b x, can be approxmated near the zero crossng by the lnear functon W b x W b x A c 3 π x (67) Near the zero crossng, the second order dervatve calculated from the nosy mage s approxmated by the lne V b x Kx M (68) where K Vb M Vb are random varables whose frst and second moments are gven by E K Wb A c 3 π V K r E ε 5 E M Wb 6b 7 π V M r E ε 3 8b 5 π (69) (7) The lne gven by (68) has the zero-crossng X M K (7) whch agan s a random varable. The probablty dstrbuton of X can be calculated explctly or numercally. Notce that the mean and the varance of X s undefned. Nevertheless the probablty dstrbuton of X can be approxmated by a normal dstrbuton N m σ wth m σ E M E K V M E K (7) Here Gauss approxmaton formulas are used together wth the fact that M and K are ndependent, and that E M accordng to (7). Combnng (7) and (7) gves 3 π σ V M E K ε 3 a b 4A b 5 (73) whch s the estmated varance of the detected edge. The approxmatons were made to obtan a smple expresson for the edge localzaton varance. Notce that (73) s only vald when a and b are farly large and when σ M E K. It s straghtforward to calculate the dstrbuton of X numercally and to get results whch are vald for a greater range n the parameters. Nevertheless, (73) s a short analytcal expresson whch descrbes the localzaton varance for a large range of the parameters. The varance s nversely proportonal to the square of the edge heght A. The varance also decreases wth ncreasng scale parameter b at low levels and ncreases at hgh levels. Ths s llustrated n Fgure. In fact the choce of scale parameter b that mnmzes the varance s b a 5. 9

20 Standard devaton Scale Parameter b Fgure : Theoretcal standard devaton of edge localzaton error versus scale parameter b at dfferent camera blur parameter a 4 5. The ntensty jump s A 5 and the standard devaton n each pxel s ε 3. The approxmatons n the calculatons are good when b s larger than 8. Smulatons and experments The whole process of mage formaton and edge detecton was smulated usng the model developed n the prevous sectons. The detected edge postons for 5 dfferent realzatons are shown n Fgure. Ths smulaton ndcates that the assumptons and the approxmatons used are reasonably vald. Prelmnary results also ndcate that sharp edges can be detected wth a standard devaton of about th of a pxel Fgure : Result after smulatng and detectng 5 one-dmensonal edges usng parameters A, a 8, b, ε. The theoretcal mean and the mddle 95% of the theoretcal dstrbuton are shown. Ths band was derved from the calculatons above. In ths smulaton 3 of 5 detected edge postons fall outsde the ths regon. The one-dmensonal analyss above can be appled drectly to horzontal and vertcal edges n a two-dmensonal mage. Each row or column s analyzed separately. Ths s llustrated n Fgure, n whch a real mage contanng a sharp edge between a lght paper and a dark background has been studed. The sub-pxel edge locaton was found for each of 7 rows, ndependently of the others. The x postons for the dfferent rows are denoted x k k 3 4. Snce we a pror know that the edge has small curvature we can use standard regresson technques to ft a low order polynomal x x k x k to the sub-pxel edge postons x k. The resdual r k x k x x k x k can then be used to estmate the standard devaton of the edge locatons. Ths was repeated for several choces of scale parameter b. Ths emprcal standard devaton wll now be compared to the theoretcal. Some of the parameters of (73) are easy to estmate, e.g. the ntensty jump A. In ths mage the jump was approxmately A

21 a b v V b c d Fgure : The fgure llustrates the applcaton of the one-dmensonal edge detector to an almost vertcal edge n a real mage of a whte paper aganst a dark background. The one-dmensonal analyss s appled to each row, 3 k 4, n the mage ndependent of the others. a: A mesh plot of the orgnal dscrete mage v. b: A contnuous nterpolaton v b s done for each row. c: For each row the x coordnate of the edge s estmated. A low order polynomal x k s ftted to the edge postons x k. usng a 6, b, A 5 and ε 3. The fgure shows the polynomal x k, the estmated postons x k wth a 95% confdence nterval for each row k. d: The fgure shows the orgnal mage and the estmated edge-ponts postons. 5. The standard devaton of mage ntensty can also be estmated n regons where the ntensty s approxmately constant. In ths mage we had ε 3. The camera blur parameter a s more dffcult to estmate. Usng a 6, a good ft s obtaned between the standard devaton computed from (73) and ts emprcal value. Ths s shown n Fgure 3. Ths strongly supports that the model and the approxmatons are vald. It s also one way to estmate the unknown parameter a n the mage acquston model. 6 Two-dmensonal edge detecton The one-dmensonal analyss can be generalzed to two dmensons n a straghtforward manner. As before mage acquston s modeled as convoluton wth kernel h followed by dscretzaton and added nose: v D h W deal e The dscrete mage s then analyzed through deal nterpolaton and smoothng: V b S b I v It s qute popular to defne edges as ponts where the gradent magntude s maxmal n the drecton of the gradent,.e. V b T V b V b (74) cf. [8]. Several smplfcatons wll be made. we wll study the stablty of edges wth respect to a gven search drecton ñ,.e. edges are defned as ponts where ñ T V b ñ (75) Usng Assumptons. and.3, the smoothed sgnal V b can be wrtten as sum of a determnstc sgnal W b and a statonary random feld E b. These parts are studed separately. Determnstc part An deal edge s modeled as x y W deal x y Aθ x (76)

22 Standard devaton Scale Parameter b Fgure 3: Edge localzaton error σ X versus scale parameter b. The curve shows the theoretcal standard devaton, cf. Equaton 73 wth a 6, A 5 and ε 3. The stars show the emprcal estmate usng resduals from a real mage at smoothng scales b 75 n steps of 5 up to b 3. Notce the close ft Fgure 4: In ths fgure we llustrate the deal ntensty functon W deal defned at all ponts, the dscrete mage w defned at nteger ponts and the scale space smoothed ntensty dstrbuton defned at all ponts W b.

23 The determnstc dscretzed mage w of ths edge s j w D h W deal j AΦ a (77) where Φ a s the one-dmensonal normal cumulatve dstrbuton functon. Below the scale-space nterpolaton W b S b I w wll be studed. W b S b h W deal S c W deal (78) where c a b. In the two-dmensonal case we estmate the edge as the locus of the ponts where the drectonal dervatve has a local maxma on a lne wth drecton ñ. If we approxmate W b as above we fnd that t s constant along the edge. Consder the dervatves n the drecton ñ cos α sn α, where α denotes the angular dfference between the search drecton ñ and the normal n to the edge. Introduce F t W b t cos α t sn α In the followng analyss we wll need F and ts frst three dervatves: F t W G c t cos α F t Acos α G c t cos α F t F t Acos t cos α α c G c t cos α Acos 3 α c t cos α c 4 G c t cos α Notce that the frst dervatve has a maxmum for t, ndependently of α, snce F and the slope of the second order dervatve at the zero crossng s (79) F Acos 3 α c 3 π (8) Random part Assume that dscrete whte nose e s added to the mage. Accordng to Theorem 4.5, the scale-space nterpolated error feld E b then s statonary wth covarance r Eb τ snc G b (8) Lemma 3. s used to smplfy the calculatons: snc G b ε 4πb e τ 4b (8) The covarance functon s the covarance between the ntensty at two postons x y and x τ x y τ y. We also need the covarance functons of the frst three drectonal dervatves of E b s t. Snce ths random feld s approxmately sotropc for large b, t s suffcent to calculate the drectonal dervatves n the s-drecton. Theorem 4.6 gves Calculatng the dervatves gves r E s s t ε π es 4b r E ss s t ε π e s r E sss s t ε π e s 4b r E s ρ r E s s t r ρ 4 r E E ss 4 s s t r ρ 6 r E E sss 6 s s t 4b 5 3b 8 3 6b 6 8b b s 6b 6 s e 3 6b 8 s 5 8b s4 t 4b 64b s4 e t 4b 56b 4 s6 e t 4b (83) (84) 3

24 The varances are gven by r E s ε 8b 4 π r E ss ε 3 6b 6 π r E ε 5 sss 3b 8 π sss have zero mean. Furthermore, E The random felds E s, E ss and E ss and E sss are ndependent, cf. []. The analyss above s only vald when we search n a drecton perpendcular to the edge. If the search drecton forms the angle α wth the edge normal, we have to evaluate r s t E and the other drectonal dervatves n s t ss τ cos α τ sn α, just as n the determnstc case. Ths gves the followng expressons for the dependences of the drectonal dervatves along the lne r E ss τ ε π e τ r E sss τ ε π e τ 4b 4b r E s τ ε π e τ Analyss of two-dmensonal edges 4b 3 6b 6 5 3b 8 8b 4 3 6b 8 τ sn α 45τ sn α 64b 6b 6 τ sn α 64b τ4 sn 4 α τ 6 sn 6 α 56b 4 5τ 4 sn 4 α 8b Smlar to the one-dmensonal case we now can calculate the dstrbuton of the edge locaton. Ths tme the second order drectonal dervatve, F x, also depends on the angle α between the search drecton and the edge normal. Close to the edge poston, F x can be approxmated by the lne y kx F x (85) (86) Acos 3 α c 3 π x (87) The second order dervatve calculated from the nosy mage s agan approxmated near the zero crossng by a lne V b x Kx M (88) where K Vb are random varables wth Ths lne (88) has the zero-crossng E K M Wb Vb Acos 3 α c 3 π V K r E r E ε 5 sss 3b 8 π E M Wb V M r E ε 3 ss 6b 6 π X (89) (9) M K (9) whch s a random varable. The probablty dstrbuton of X can be approxmated by the normal dstrbuton N m σ wth m σ E M E K V M E K (9) 4

25 Fgure 5: Resultng resdual hstogram for smulated data. The theoretcal standard devaton of 499 agrees wth the expermentally estmated of t τ α 5 n s Fgure 6: The fgure llustrates some notatons used n the analyss of two-dmensonal edge detecton. where Gauss approxmaton formulas are used together wth the fact that M and K are ndependent, and E M. Combnng (9) and (9) gves V X V M E K ε 3 a b 8A b 6 cos 6 α 3 (93) Ths s the estmated varance of the detected edge. Observe that the varance decreases wth ncreasng heght, A, of the edge. The varance also ncreases when α ncreases, that s when we do not search perpendcularly to the lne. The detected edge as a random process Let the true edge γ be parametrzed by curve parameter τ. Apply the edge detector n search drecton ñ from every pont γ τ. The detected edge can then be parametrzed as γ τ γ τ z τ ñ, where z descrbes the devaton of the detected edge from the true edge. The devaton z τ can be approxmated as z τ V b ss γ τ W b sss γ τ (94) 5

26 .5.5 x Fgure 7: Results from two-dmensonal edge detecton wth smulated data. Left: Edge poston errors n a drecton roughly perpendcular to the edge at dfferent postons along the edge. The result from four smulatons are shown. Dstant errors are not correlated but there s a hgh correlaton between edge poston errors at spatally close postons. The resduals can be modeled as samples of a random process wth respect to the parametrzaton of the curve. Rght: Theoretcal and estmated covarance functons for the resdual error process. where W b sss γ τ Then the covarance between the devatons z τ and z τ s where C z τ z τ C V b ss γ τ V b Acos 3 α c 3 π n (86). Hence, z s a statonary process wth covarance functon C V b ss γ τ V b ss γ τ c 6 π A cos 6 α (95) ss γ τ r E sss τ τ snα τ τ cosα r z τ r τsn α E τcos α F ss ε c τ 6 A cos 6 α e 4b 3 6b 6 3 6b 8 τsn α 64b τsn α 4 (96) Ths could be used to extract the mean value of the random process related to the lne. The mean value can be used as the estmated locaton of the lne. If we assume that the search drecton dffers at most 5 degrees from the perpendcular drecton to the edge we get the followng estmate of the covarance functon r z τ ε 3 a b 3 τ 8A b 6 e 4b (97) Notce that the parameter τ s measured as the arclength along the lne. Snce the edge s detected as the soluton to the equaton Wb we can regard the edge as a level set to W. Ths makes t possble to use a more refned analyss than the approxmaton wth the tangent lne descrbed above. Ths s dscussed n detal n [9]. Implementaton and experments The two-dmensonal edge detector descrbed above has been mplemented. Its performance on both smulated and real mages have been nvestgated. In the smulatons the true edge was well defned. The devatons z were studed both wth respect to dfferent realzatons but also as a random process along the edge. Fgure 4 llustrates the orgnal ntensty W, the dscrete measured ntensty w and the smoothed mage ntensty W b. The dscrete mage was dsturbed wth smulated Gaussan uncorrelated nose. Ths mage was then used n the edge detecton routnes to calculate the edges along lnes roughly perpendcular to the true edge. In these smulatons the search lne was radans off the normal. Ths was done at several postons along the 6

27 a b c. 4 x Low curvature Fgure 8: Implementaton on real mage. Along curved parts wth small curvature, the assumpton that the curve s approxmately lnear holds. The resdual after fttng a low degree polynomal to the edge s an approxmately random process along the curve. The emprcal covarance functon s reasonably close to the theoretcal one. v v Fgure 9: Two real mages of a hghly textured floor. These are used to llustrate correlaton. true edge. Fgure 5 shows a hstogram of the resduals at one poston for several realsatons. The theoretcal and emprcal standard devatons agree well. The fgure also shows the edge postons wth confdence ntervals. Fgure 7 llustrates four dfferent realzatons of the devaton z n search drecton, from the true edge as we move along the edge. The devaton z τ s a statonary random processes. Edge poston errors at dstance τ have covarance r z τ. The theoretcal and emprcal covarance functons agree well as can be seen n the fgure. The edge detector was also appled to a real mage of a dark object aganst a lght background, cf. Fgure 8.a. Part of the contour s smooth and has small curvature. The extracted curve at ths patch was analyzed and the resduals along the curve were estmated n the followng way. The curved part, approxmately pxels long, was rotated to become almost horzontal. The orgnal object, whch had been drawn on a hgh resoluton laser prnter, was known to have very small varatons n curvature. Ths motvated fttng a thrd order polynomal to the curve. The resdual s shown n Fgure 8.b, and defnes a statonary random process. The emprcal covarance functon was estmated from the resduals usng standard technques. Fgure 8.c shows the resultng emprcal covarance functon together wth the covarance computed from the model (96) usng the followng parameters: A 7, a 7, b, ε 7, and α 3 degrees. The theory agrees well wth the emprcal results n ths case. Our analyss was based on deal, straght step dscontnutes. In ths case the edge detector s unbased. Bas s expected for more realstc stuatons and at edge ponts wth consderable curvature. 7 Sub-pxel correlaton Analyss of sub-pxel correlaton s another applcaton of the theory presented here theory. Correlaton s usually done on pxel level, where a regons of one mage s translated n whole pxel unts and matched to parts of a second mage so that the sum of squared dfferences are mnmzed. The result after least squares mnmzaton n a typcal case s llustrated n Fgure. The stochastc errors of pxel correlaton s dffcult to analyze, manly because the translaton between the regons n the two mages usually s of sub-pxel type. 7

28 v v v v Fgure : Regons n two mages are correlated wth pxel translatons usng least squares of the resduals V b V b V b V b Fgure : Regons n two mages are correlated wth sub-pxel translatons usng least squares of the resduals of the restored contnuous scale-space representaton at scale b 9. The accuracy of the sub-pxel translaton can be analyzed through stochastc models of the resdual feld. The resdual W W can also be used to emprcally estmate the stochastc propertes of the error feld E b. The gray level scale s dfferent for the resdual W W. A substantal mprovement s obtaned by usng scale-space restoraton of contnuous mages. Ths makes t possble to correlate regons n two mages wth sub-pxel translatons wth much hgher precson than obtaned by ordnary methods. Furthermore, a proper modelng of the resdual feld makes t possble to analyze the stochastc propertes of the localzaton error. The dea s that, at least locally, the mages only dffer by an unknown translaton h. Denote by V W E and V W Ē the restored ntensty felds n two mages for a fxed scale b. The determnstc functons are dentcal except for a translaton. For a fxed translaton h h h, we thus have W t W t h t To determne the translaton h wth sub-pxel accuracy a least squares ntegral s mnmzed, F h V t V t h dt t Ω The result of such a mnmzaton s shown n Fgure. Furthermore, the resdual feld V t V t h can be used to emprcally study the stochastc propertes of the camera nose e. Statstcal analyss The qualty of the estmated sub-pxel translaton, ĥ argmnf h can be analyzed usng the statstcal model gven above. Let X ĥ h be the error n estmated translaton. Wthout loss of generalty we can assume that h s zero. The stochastc varable X s a result of mnmsng the functon F, whch n turn contans the random felds E andē. What s the the mean and covarance matrx of X? Ideally the mean should be zero, ndcatng that ĥ s a non-based estmate of h, and the covarance should hopefully be small. Lnearsng F around the true dsplacement h, we fnd that the squared resdual F can be approxmated as F X X T AX bx f (98) 8